$\ell^p$-Stability of Weighted Persistence Diagrams
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908321933426688 |
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| author | Gülen, Aziz Burak Mémoli, Facundo Patel, Amit |
| author_facet | Gülen, Aziz Burak Mémoli, Facundo Patel, Amit |
| contents | We introduce the concept of weighted persistence diagrams and develop a functorial pipeline for constructing them from finite metric measure spaces. This builds upon an existing functorial framework for generating classical persistence diagrams from finite pseudo-metric spaces. To quantify differences between weighted persistence diagrams, we define the $p$-edit distance for $p\in [1,\infty]$, and-focusing on the weighted Vietoris-Rips filtration-we establish that these diagrams are stable with respect to the $p$-Gromov-Wasserstein distance as a direct consequence of functoriality. In addition, we present an Optimal Transport-inspired formulation of the $p$-edit distance, enhancing its conceptual clarity. Finally, we explore the discriminative power of weighted persistence diagrams, demonstrating advantages over their unweighted counterparts. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_11694 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\ell^p$-Stability of Weighted Persistence Diagrams Gülen, Aziz Burak Mémoli, Facundo Patel, Amit Algebraic Topology We introduce the concept of weighted persistence diagrams and develop a functorial pipeline for constructing them from finite metric measure spaces. This builds upon an existing functorial framework for generating classical persistence diagrams from finite pseudo-metric spaces. To quantify differences between weighted persistence diagrams, we define the $p$-edit distance for $p\in [1,\infty]$, and-focusing on the weighted Vietoris-Rips filtration-we establish that these diagrams are stable with respect to the $p$-Gromov-Wasserstein distance as a direct consequence of functoriality. In addition, we present an Optimal Transport-inspired formulation of the $p$-edit distance, enhancing its conceptual clarity. Finally, we explore the discriminative power of weighted persistence diagrams, demonstrating advantages over their unweighted counterparts. |
| title | $\ell^p$-Stability of Weighted Persistence Diagrams |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2504.11694 |